Finite Group DatabasebetaFinite Group Databasebeta| order | 1536 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
| direct product | C23×#5936 |
| derived | C22⋊A4 ⋊ C23×C4 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C24 | #1020 |
| commutator, hypocenter | C22⋊A4 | C23×C4 |
| Frattini | C2 | C23×#1020 |
| Fitting, socle | C28 | D3 |
| prime radical | C24×C22⋊A4 | C2 |
| cosocle | C2×C22⋊A4 | C24 |
| LCS | G ▷ C22⋊A4 | |
| UCS | C1 ◁ C24 | |
86869 conjugacy classes of subgroups; only the 453 normal subgroups are shown.
This group has 86869 conjugacy classes of subgroups — too many to draw. Only its 453 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 96 |
A minimal faithful permutation representation — 4 generators on 18 points:
(5,6)(7,8,10)(12,13,14);(3,4)(11,14)(12,13);(8,9)(13,14)(15,16,17,18);(1,2)160 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#72246416 |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((5,6)(7,8,10)(12,13,14), (3,4)(11,14)(12,13), (8,9)(13,14)(15,16,17,18), (1,2));